Exercise 3-2 from Calculus on Manifolds by Spivak:
Let $A\subset R^n,\ f:A\rightarrow R$ an integrable (in the sense of Darboux) function. Let $g=f$ except at finitely many points. Prove that $g$ is also integrable and $\int_Af=\int_A g$.
Note that there is a similar question show that $g$ is integrable, with $f=g$ except in finite set and $f$ integrable but the answers assume the knowledge of measure theory whereas Spivak doesn't.
I guess I need to use the criterion saying that $f$ is integrable iff there is a partition $P$ of $A$ such that $U(f,P)-L(f,P)< \epsilon$ for any $\epsilon < 0$. But I don't know how to apply it to both functions. I thought about considering $f-g$ (which should be integrable except finitely many points), but Spivak doesn't even state that the sum of two integrable functions is integrable, so perhaps I'm not supposed to use this. (Even if I consider $f-g$, I don't know how to proceed).